The operator <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_2132_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="115" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mi>M</mi> </msup> <msup> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">/</mo> <mi>d</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mi>M</mi> </mrow> </msup> </math></EquationSource> <EquationSource Format="TEX">$(-1)^{M} (d/dx)^{2M}$</EquationSource> </InlineEquation>, with the factor <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_2132_Article_IEq2.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mi>M</mi> </msup> </math></EquationSource> <EquationSource Format="TEX">$(-1)^{M}$</EquationSource> </InlineEquation> chosen to ensure the positivity of associated Green’s function, plays a central role in the formulation. An explicit integral representation of Green’s function is derived, avoiding the appearance of unknown boundary derivatives and ensuring symmetry and strict positivity.This positivity is rigorously proven through a new integral formula, based on a symmetry-preserving change of variables and the use of Euler polynomials. The normalization constant <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_2132_Article_IEq3.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msub> <mi>κ</mi> <mi>M</mi> </msub> </math></EquationSource> <EquationSource Format="TEX">$\kappa _{M}$</EquationSource> </InlineEquation> is defined as a factorial Hankel determinant and evaluated using orthogonal polynomials associated with a non-standard inner product.These results form a new analytic framework for constructing Green’s functions and evaluating determinant structures, with potential applications to the determination of sharp constants in Sobolev-type inequalities and the design of positive definite kernels for approximation theory and machine learning.

Strict positivity and explicit integral representation of Green’s function for high-order clamped boundary value problems

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2025年9月26日

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Kazuo Takemura,
Atsushi Nagai,
Hiroto Sekido

摘要

The operator ${(- 1)}^{M} {(d / d x)}^{2 M}$ $(-1)^{M} (d/dx)^{2M}$ , with the factor ${(- 1)}^{M}$ $(-1)^{M}$ chosen to ensure the positivity of associated Green’s function, plays a central role in the formulation. An explicit integral representation of Green’s function is derived, avoiding the appearance of unknown boundary derivatives and ensuring symmetry and strict positivity.

This positivity is rigorously proven through a new integral formula, based on a symmetry-preserving change of variables and the use of Euler polynomials. The normalization constant $κ_{M}$ $\kappa _{M}$ is defined as a factorial Hankel determinant and evaluated using orthogonal polynomials associated with a non-standard inner product.

These results form a new analytic framework for constructing Green’s functions and evaluating determinant structures, with potential applications to the determination of sharp constants in Sobolev-type inequalities and the design of positive definite kernels for approximation theory and machine learning.